Review Notes
I. ALGEBRA
1.1 Laws/ Properties of Exponents
E1.
E5.
E2.
if
E6.
if
E7.
if
E8.
E3.
E9.
if
E4
1.2 Laws/ Properties of Radicals
R1.
R4.
R2.
R5.
R3.
R6.
1.3 Special Product
S1.
S5.
S2.
S6.
S3.
S7.
S4.
1.4 Factorization Patterns
F1.
F2.
F3.
F4.
for any positive integer
F5.
for any positive odd integer
1.5 Quadratic Equation
a) General Form:
where
coefficient of the 2nd-degree term
coefficient of the 1st-degree term
constant term
b) Quadratic Formula
c) Sum of the Roots
e) Nature of the Roots
1. If
2. If
3. If
d. Product of the Roots
, then
, then
, then
and
and
and
are real and unequal (or distinct).
are real and equal.
are complex and unequal.
1
1.6 Remainder Theorem
For any constant , if a polynomial
is divided by
2
, the remainder
is equal to
.
1.7 Factor Theorem
For any polynomial
and any number ,
1.8 Ration and Proportion
a) Notation for ratio:
if and only if
is equal to zero.
or
b) Notation for proportion:
c) If
is a factor of
or
, then
1.
d)
2.
3.
4.
5.
3.
4.
or
where:
is equivalent to
1.
2.
1.9 Variation
a) Direct variation
b) Inverse variation
c) Joint variation
constant of proportionality
1.10 Binomial Expansion
a) Binomial Theorem
B1.
b) rth term
B2.
rth term =
1.11 Arithmetic Progression (AP)
a) Def: An arithmetic progression is a sequence of numbers such that successive numbers differ by a constant
b) Formulas:
A1.
where
1st term
common difference
A2.
nth or last term
A3.
number of terms
sum of the first n terms
c) Arithmetic Mean (AM) of two numbers
A4.
and
d. Arithmetic Mean of
AM
A5.
AM
1.12 Geometric Progression (GP)
a) Def: A geometric progression is a sequence of numbers such that the same quotient is obtained by dividing a
term by the preceding term.
b) Formulas:
G1.
where
1st term
number of terms
G2.
nth or last term
G3.
sum of the first
c) Geometric Mean (GM) of two numbers and
G4.
GM
if
G5.
3
GM
common ratio
terms
if
d) Geometric Mean of
G6.
e) Sum of Infinite Progression
GM
G7.
1.13 Harmonic Progression (HP)
a) Def: A harmonic progression is a sequence of numbers such that the reciprocals of the numbers form an
arithmetic progression.
b) rth term of a harmonic progression
To find the rth term of a harmonic progression, write the equivalent arithmetic progression by
taking the reciprocal of the given terms. Use the formulas for the AP to find the rth term corresponding to the
rth term of the HP. The rth term of the HP equals the reciprocal of the rth term of the AP. In symbol,
c) Harmonic Mean (HM) between two numbers
H1.
d) Harmonic Mean of
HM
H2.
1.14 Complex Numbers
a) Cartesian notation:
where
b)
c)
d)
e)
f)
and
HM
real part
imaginary part
imaginary unit
Magnitude:
Conjugate complex numbers:
The sum of two conjugate complex numbers:
Difference of two conjugate complex numbers:
Product of two conjugate complex numbers:
and
g) Quotient of two conjugate complex numbers:
h) Fundamental Operations
Sum
:
Difference
:
Product
:
Quotient
:
1.15 Partial Fractions
a) Def:
1. Proper Fraction – an algebraic fraction in which the degree of the numerator is less than the degree
denominator.
2. Partial Fractions – the simpler fractions to which the proper fraction has been resolved.
b) Methods of resolving proper fraction:
A proper fraction of the form
where
and
are polynomials with integral coefficients, can
be resolved into partial fractions, subject to the following cases:
Case I:
When the factors of
are linear and none is repeated.
Rule: To each non-repeated linear factor
, there corresponds a partial fraction of the form.
Case II:
When the factors of
are linear and some are repeated.
Rule: To each repeated linear factor
, there corresponds the sum of
of the form
Case III:
partial fractions
When the factors of
are irreducible quadratic factors and none is repeated.
Rule: To each non-repeated irreducible quadratic factor
, there corresponds a
partial fraction of the form
4
Case IV:
where and are constants to be determined.
When the factors of
are irreducible quadratic factors and some are repeated.
Rule: To each repeated irreducible quadratic factor
, there corresponds the
sum of partial fractions of the form
where
are constants to be determined.
II. PROBABILITY
1.
Fundamental Principle of Counting (FPC)
If one thing can be accomplished in n1 different ways, a second thing can be accomplished in n2 different ways, …,
and finally a kth thing can be accomplished in nk different ways, then k things can be accomplished in
number of ways.
Examples
a. In how many ways can three tossed coins fall?
b. In how many ways can two rolled dice land?
c. How many possible outcomes are there when a coin is tossed once or a die is rolled once?
2.
Permutation
Def: 2.1 A permutation is an arrangement of a set of objects in a definite order.
Example
Consider the set J = {a, b, c}.What is the number of permutations of the elements of set J?
2. 1 Linear Permutation
Suppose that we are given n distinct objects and wish to arrange r of these objects in a line. Since there
are n ways of choosing the 1st object, n – 1 ways of choosing the 2nd object, … , and finally n – r +1 ways of
choosing the rth object, it follows by the FPC that the number of different arrangements or permutations is given by
n (n – 1) (n – 2) … (n – r + 1)
We call nPr the number of permutations of n objects taken r at a time. If n = r, then
nPr = n (n – 1) (n – 2) … 1 = n!
If n objects are to be arranged taken r objects at a time, then the number of distinct arrangements is given
by the formula:
nPr
Examples:
a. In how many ways can 7 different books be arranged in a shelf?
b. In how many ways may the first, second, and third prizes be drawn from 8 lottery tickets?
2.2 Circular Permutation
If n objects are to be arranged in a circular manner, then the number of distinct arrangements is (n – 1)!
Example:
In how many ways can 5 people be arranged in a circular table?
2.3 Permutation with Repetition
The number of distinct permutations of n things of which n1 are of one kind, n2 of a second kind, … , nk of
kth kind is
Example :
In how many different ways can 5 red, 3 blue, and 4 green bulbs be arranged in a wire of a
Christmas tree lights with 12 sockets?
3.
Combination
What distinguishes permutation from combination is order. That is, a combination is also an arrangement of a set of
objects but without regard to order. Hence, changing the order of any combination does not make a new combination.
The number of combinations of n distinct objects taken r at a time is
Examples
a. How many combinations of two letters are possible using the letters a, b, and c.
b. In how many ways can a committee of 5 people be chosen from 9 people?
5
4. Probability of Events
Def: 4.1 Probability is the likelihood of occurrence of an event.
Def: 4.2 Experiment is an activity that’s under consideration and which can be done repeatedly.
Examples
a. Drawing a card from an ordinary deck of 52 cards
b. Tossing a coin
c. Rolling a die
Def: 4.3 Sample space is the set of all possible outcomes of an experiment.
Examples
a. In tossing a coin, the possible outcomes are head or tail. Hence the sample space is S = {H, T}.
b. In rolling a die, the sample space is S = {1, 2, 3, 4, 5, 6}
Def: 4.4 Sample point is an element of the sample space. In tossing a coin, there are two sample points: head and tail.
Def: 4.5 An event is any subset of the sample space. A simple even is one that consists of exactly one outcome,
hence it cannot be decomposed. An event is compound if it consists of more than one outcome.
Def: 4.6 The complement of an event A with respect to the sample space, S is the set of all elements of S that are not
in A (denoted by A’)
4.1 Theoretical Probability – the probability of an event E denoted by P(E) is given by
where n(E) = # of favorable outcomes and n(S) = # of possible outcomes
If E is any event, then the probability of an event denoted by P(E) has a value between 0 and 1 inclusive. In symbol
If P(E) = 1, then E is sure to happen. If P(E) = 0, then E is impossible to happen. Moreover, if the
probability that E will not happen is P(E’), then P(E) + P(E’) = 1.
Examples
a. Consider the experiment of rolling a die. What is the probability of getting a prime number?
b. If the probability that Jovie will graduate is , what is the probability that she will not graduate?
4.2 Conditional Probability
The conditional probability of B, given that A has occurred, is denoted by P(B/A). Since A known to
have occurred, then A becomes the new sample space replacing the original sample space S. Below is the formula
for the conditional probability of B given A.
if P(A) ≠ 0
Example
Find the probability that a single toss of a die resulted in a number less than 4 if the toss resulted in even number.
4.3 Mutually and Non-mutually Exclusive Events (The Additive Rule)
Def: 4.3.1 Two events are mutually exclusive if they cannot happen at the same time. Otherwise they are said to
be non-mutually exclusive.
If A and B are any two events, then the probability that A or B will happen is given by the formula:
P(A or B) = P(A B) = P(A) + P(B) – P(A B)
where P(A B) is the probability that both A and b will happen.
If A and B are mutually exclusive, then P(A B) = P(A) + P(B). Because if A and B are mutually exclusive
the probability of their joint occurrence is zero.
Example:
A card is drawn from a standard deck. What is the probability of getting a) an ace or a king? b) a red or a
face card?
4.4 Dependent and Independent Events (The Multiplication Rule)
Def: 4.4.1 Two events are independent if the occurrence of one does not affect the occurrence of the other.
Otherwise they are said to be dependent.
If A and B are any two events, then the probability that A and b will happen is given by the formula:
P(A and B) = P(A B) = P(A) P(B/A)
Where P(B/A) is the probability that A will happen given that A happened already. However, if A and B are
independent events then P(A B) = P(A) P(B) since P(B/A) = P(B)
Example:
Two cards are drawn one at a time from a well-shuffled deck of cards. Find the probability that they are
both kings if the first card is (a) replaced, (b) not replaced.
5. Binomial Distribution
Def: 5.1 An experiment that has only two possible outcomes, for instance a success and a failure, is called a binomial
experiment.
6
I.
II.
III.
IV.
Properties of a binomial experiment
The experiment consists of repeated trials.
Each trial results in a outcome that can be classified as a success or a failure.
The probability of success remains constant from trial to trial.
The repeated trials are independent.
Def: 5.2 If a binomial experiment can result in success with probability p and failure with probability q = 1 – p, then the
probability distribution of the binomial random variable x, the number of successes in n independent trials, is
b(x; n, p) =
px qn-x
Example
The probability that a certain kind of component will survive a given shock test is , find the probability that exactly 2
of the next 4 components tested survive.
III. PLANE TRIGONOMETRY
I.
TRIGONOMETRIC FUNCTION VALUES
A. Trigonometric Functions of Angles Between 0° and 90°
In the right triangle shown below, the trigonometric functions for
side
opposite θ
are defined as follows:
90° - θ
hypotenuse
θ
side adjacent to θ
You can also write,
Thus,
Example 1:
1.
2.
B. Values of Trigonometric Functions of Special Angles
You can easily remember the values of the trigonometric fucntions sine, cosine and tangent by using the special
triangle 30°-60°-90° and 45°-45°-90° and by applying the Pythagorean Theorem and the definition cited earlier.
2
60°
1
1
30°
1
There are other trigonometric ratios from the sine, cosine and tangent. These are secant, cosecant and cotangent.
They are defined as follows:
7
The table below shows the values of the six trigonometric functions of special angles.
2
1
1
2
II.
TRIGONOMETRIC IDENTITIES
A. Reciprocal Identities
B. Pythagorean Identities
C. Quotient Identities
D. Co – Function Identities
E. Even – Odd Identities
F. Sum – Difference Formulas
G. Double Angle Formulas
H. Power – Reducing/Half Angle Formulas
8
I.
Sum – to – Product Formulas
J. Product – to –Sum Formulas
III.
LAW OF SINES AND LAW OF COSINES
Right triangle trigonometry can be used to solve problems involving right triangles. However, many interesting
problems involve non – right or oblique triangles. The law of sines and the law of cosines can be used to solve these types
of problems as well as those involving right triangles.
A
A. Law of Sines
This form is easier to use when finding an unknown side.
The law of sines can also be written as:
b
c
This form is easier to use when solving for an unknown angle.
C a bearing of 285° from A. B
Example 1: A, B and C are points on a plane. B is 6 km due west of A. C is 5 km from B with
a
What is the measure of
?
A
B. Law of Cosines
b
c
This form is easier to use when finding an unknown side.
This form is easier to use when finding an unknown angle.
C
B
a
Example 2: A ship sails from a port (P), due west of a lighthouse (L) 6 km away. The ship then sails 10 km to an island
(A) with a bearing of 30°. Find (a) Distance AP. (b) The bearing of P from A.
C. Law of Tangent
a.
b.
c.
D. Choosing which of the Law of Sines or Law of Cosines to Use
 The law of sines can be used to determine the measures of missing angles or sides of triangles when the
measures of two angles and a side (AAS or ASA), or of two sides and a non-included angle (SSA), are known.
 The law of cosines can be used when the measures of two sides and an included angle (SAS) or of the three sides
are known (SSS).
IV. PLANE AND SOLID GEOMETRY
I.
TRIANGLES
A. Some theorems about Triangles
1. Isosceles Triangle Theorem. If two sides of a triangle are congruent, then the angles opposite these sides are
congruent. Conversely, if two angles of a triangle are congruent, then the sides opposite them are congruent.
2. Every equilateral triangle is equiangular and conversely.
3. The sum of the measures of the angles of a triangle is 180.
B.
Theorems on Right Triangles
1. Pythagorean Theorem
In a right triangle, the square of the length of the hypotenuse is
equal to the sum of the squares of the lengths of the legs.
9
a
c
b
2. In a 30 – 60 – 90 triangle,
a. The hypotenuse us twice as long as the shorter leg (the leg
Opposite the 30° angle), and
b. The longer leg is
times as long as the shorter leg.
60
a
2a
30
a
3. In a 45 – 45 – 90 triangle, the hypotenuse is
times as long as either leg.
45
a
a
4. Angles Outside the Triangle
The measure of an exterior angle of a triangle is equal to
the sum of the measures of the remote interior angles.
a
45
2
1
5. Exterior Angle Inequality Theorem
The measure of an exterior angle of a triangle is greater
than the measures of either remote interior angle. At the right,
3
4
3
and
4
2
1
6. Triangle Inequality Theorem
The sum of the lengths of the two shorter sides of a triangle is
greater than the length of the third side.
a
a+b>c
c
b
TRIANGLE CONGRUENCE POSTULATES
Given two triangles. There are four ways to show that they are congruent using only three pairs of corresponding
congruent parts:
SAS Congruence Postulate. If two sides and the included angle of one triangle are congruent respectively to the
corresponding two sides and the include angle of another triangle, then the two triangles are congruent.
ASA Congruence Postulate. If two angles and the included side of one triangle are congruent respectively to the
corresponding two angles and the included side of another triangle, then the two triangles are congruent.
SSS Congruence Postulate. If the three sides of one triangle are congruent respectively to the corresponding three sides
of another triangle, then the two triangles are congruent.
SAA Congruence Theorem. If a side and two angles adjacent angles of one triangle are congruent respectively t the
corresponding side and two adjacent angles of another triangle, then the two triangles are congruent.
Corresponding Parts Principle. If two triangles are congruent by SAS, ASA, SSS, or SAA, then their remaining
corresponding parts are also congruent.
Examples:
Each pair of marked triangles are congruent by the indicated congruence postulate.
60
60
32
32
ASA
SAS
SSS
Triangle Congruence for Right Triangles
From the triangle congruence postulates, any two triangles may be congruent by any of the following principles:
LL Congruence. Two right triangles are congruent if the two legs of one are congruent, respectively to the corresponding
two legs of the other. (By SAS)
LA Congruence. Two rights triangles are congruent if a leg and an adjacent acute angle of one are congruent, respectively,
to the corresponding leg and an adjacent acute angle of the other. (By ASA; by SAA if the acute angles are not adjacent)
10
HL Congruence. Two right triangles are congruent if the hypotenuse and a leg of one are congruent, respectively, to the
corresponding hypotenuse and a leg of the other. (By Transitivity)
LL
LA
HL
First Minimum Theorem
The shortest segment joining a point to a line is the perpendicular segment. Thus, the distance between a line and
an external point is the length of the perpendicular segment from the point to the line.
Similar Triangles
Two triangles are said to be similar if
a. Their corresponding angles are congruent, and
b. Their corresponding sides are proportional.
Examples of similar triangles:
Similarity Postulates:
1. AA Similarity. If two angles of one triangle are congruent to two corresponding angles of another triangle, then the
triangles are similar.
2. SAS Similarity. If an angle of one triangle is congruent to a corresponding angle of another triangle and the sides
that include these angles are proportional then the triangles are similar.
3. SSS Similarity. If all the three sides of one triangle are proportional to the lengths of the corresponding sides of
another triangle, then the triangles are similar.
4. Midsegment Theorem for Triangles. A segment whose endpoints are the midpoints of two sides of a triangle is
4.1 Parallel to the third side, and
A
4.2 Half the length of the third side.
D
E
B
C
and
A
5. Side – Splitting Theorem. If a line parallel to a side of a triangle
intersects the other two sides in distinct points, then it cuts off
segments which are proportional to these sides.
D
B
E
C
6. Similarity in a right triangle. The altitude to the hypotenuse of a right triangle forms two triangles that are each
similar to the original triangle and to each other.
7. Given a right triangle and the altitude to the hypotenuse. (a) The altitude to the hypotenuse is the geometric
mean of the segments into which it separates the hypotenuse. (b) Each leg is the geometric mean of the
hypotenuse and the segment adjacent to the leg.
Thus, in the marked adjoining figure,
C
. Thus,
A
D
11
B
Hence,
.
Moreover,
. Hence,
. Finally,
, hence
Examples:
1. Two angles of
have measures, 45° and 15°, while two angles of
the triangles similar? By what similarity theorem of definition?
a. Solution: Yes, by the AAA Similarity Theorem
have measures 120° and 45°. Are
2. One angle of
measures 40° and the sides that include the angle measures 5 cm each. Another triangle has
an angle that measures 70° and the sides that include these angle measures 8 cm each. Are the triangles similar?
a. Solution: Yes, by the SAS Similarity Theorem
3. Given the figure as marked. Find PS.
S
Q
15
P
x
x
C
12
9
R
T
4. In the marked figure at the right,
and
. Find
.
Solution:
A
B
D
QUADRILATERALS
A quadrilateral is a four – sided figure.




If all four angles of a quadrilateral are right angles, then the quadrilateral is a rectangle.
If all four angles of a quadrilateral are right angles, and all four sides are congruent, then the quadrilateral is a
square.
If both pairs of opposite sides are parallel, then the quadrilateral is a parallelogram.
If one and only one pair of opposite sides are parallel, then the quadrilateral is a trapezoid.
Rectangle
Square
Parallelogram
12
Trapezoid
Isosceles Trapezoid
THEOREMS ON QUADRILATERALS
1. Each diagonal separates a parallelogram into two congruent triangles.
2. In a parallelogram, any two opposite sides are congruent.
Corollary: If two lines are parallel, then all points of each line are equidistant from the other line.
Recall: The distance between a line and an external point is the length of the perpendicular
segment from the point to the line.
The distance between any two parallel lines is the distance from any point of one to the other.
3. In a parallelogram, any two opposite angles are congruent.
4. In a parallelogram, any two consecutive angles are supplementary.
5. The diagonals of a parallelogram bisect each other.
Application:
1. The segment joining the midpoints of two sides of a triangle is parallel to the third side and half as long.
2. A rhombus is a parallelogram all of whose sides are congruent.
3. A rectangle is a parallelogram all of whose angles are congruent.
4. A square is a rectangle all of whose sides are congruent.
5. If a parallelogram has one right angle, then it is a rectangle.
6. In a rhombus, the diagonals are perpendicular bisectors to each other.
Rhombus
7. If the diagonals of a quadrilateral bisect each other and are perpendicular, then the quadrilateral is a rhombus.
CIRCLES
Definition
Let P be a point in a given plane, and r be a positive number, the
circle with center P and radius is the set of all points of the plane whose
distance from P is r.
B
A
P
SPHERE
Definition
Let P be a point and let r be a positive number. The sphere with center
P and radius r is the set of all points of space whose distance from P is r.
R
A
P
B
Basic Terms on Circles and Spheres








two or more spheres or two or more circles with the same center are called concentric.
R
A chord of a circle is a segment whose endpoints line on the circle.
A line which intersects a circle in two points is called a secant of the circle.
A
B
A chord of a sphere is a segment whose endpoints lie on the sphere.
A diameter of a circle or sphere is a chord containing the center.
A radius of a circle or a sphere is a segment from the center to a point of the sphere.
The interior of a circle is the set of all points of the plane whose distance from the center is less than the radius.
The exterior of a circle is the set of all points of the plane whose distance from the center is greater than the radius.
Definition
A tangent to a circle is a line (in the same plane) which intersects
the circle in one and only point. This point is called the point of tangency.
R
A
B
Theorems Circles and Spheres
1. The intersection of a sphere with a plane through its center is a circle with the same center and the same
radius.
2. The intersection of a sphere with a plane through its center is called a great circle of the sphere.
3. A line perpendicular to a radius at its outer end is tangent to the circle.
4. Every tangent to a circle is perpendicular to the radius drawn to the point of tangency.
5. The perpendicular from the center of a circle to a chord bisects the chord.
6. The segment from the center of a circle to the midpoint of a chord which is not a diameter is perpendicular to
the chord.
7. In the plane of a circle, the perpendicular bisector of a chord passes through the center.
8. In the same circle or in congruent circles, chords equidistant from the center are congruent.
9. In the same circle or in congruent circles, any two congruent chords are equidistant from the center.
13
10. If the line and the circle are coplanar, and line passes the interior of the circle, then it intersects the circle in two
and only two points.
Definition
Two circles are tangent if they are tangent to the same line at the same point. If two tangent circles are coplanar,
and their centers are on the same side of their common tangent, then they are internally tangent. If two tangent circles are
coplanar, and their centers are on opposite sides of their common tangent, then they are externally tangent.
Internally tangent
Externally tangent
ARC OF CIRCLES
A
In the adjoining circle at the right, P is the center. The set of points
(darkened) on the circle in the interior of
is the minor arc
. The
remaining set of points on the circle is the major arc
. A and B are the
endpoints of the arcs.
P
B
Definition
A central angle of a circle is an angle whose vertex is the center of the circle.
In the figure,
is a central angle.
A
P
B
Definition
1. The degree measure of an arc is the measure of the corresponding central angle.
2. The degree measure of a semicircle is 180
INSCRIBED ANGLES AND INTERCEPTED ARCS
Definition
An angle is inscribed in an arc if
1. The sides of the angle contain end points of the arc and
2. The vertex of the angle is a point on the circle, but not an end point, of the arc.
Definition
An angle intercepts an arc if
1. The endpoints of the arc lie on the angle,
2. All other points of the arc are in the interior of the angle, and
3. Each side of the angle contains an end point of the arc.
Theorems on Inscribed Angles and Intercepted Arcs
1. The measure of an inscribed angle is half the measure of the intercepted arc.
2. An angle inscribed in a semicircle is a right angle.
3. Every two angles inscribed in the same arc are congruent.
Definitions
A triangle is inscribed in a circle if the vertices of the triangle lie on the circle. If each side of the triangle is tangent
to the circle, then the triangle is circumscribed about the circle.
A quadrilateral is inscribed in a circle if the vertices of the quadrilateral lie on the circle. If each side of the
quadrilateral is tangent to the circle, then the quadrilateral is circumscribed about the circle.
A polygon is tangent to a circle, then the polygon is circumscribed about the circle.
Inscribed Triangle
Circumscribed Triangle
Circumscribed Quadrilateral
14
Inscribed Quadrilateral
SOLIDS AND THEIR VOLUMES
Right Prisms
Cube
Triangular Prism
Pyramid
Geometry FORMULAS for PERIMETER (P), CIRCUMFERENCE (C), AREA (A), SURFACE AREA (SA), and VOLUME (V)
Rectangle
Right Rectangular Prism
P = 2a + 2b
A = ab
b
V = abc
S = 2(ab + ac + bc)2
c
a
b
a
Square
Cube
s
P = 4s
A = s2
s
V = s3
S = 6s2
s
Triangle
s
s
Right Prism
P=a+b+c
A = ab
a
c
V = Ah
S = 2A + Ph
h
Where A = area of base
and P = perimeter of
base
b
Parallelogram
A
P
Right Regular Pyramid
P = 2a + 2b
V = Ah
a
h
A = bh
e
S = A + Pl
A
b
Trapezoid
Right Circular Cylinder
b
P=a+b+c+d
A = (a + b)h
c
P
h
r
V = Ah
= r2h
d
h
S = A + Ch
a
A
C
C
= 2 r2 + 2 rh
Regular n – gon
P = ns
Right Circular Cone
s
V = Ah
r
A = rP
h
l
S = A + Pl
A
r
C
Circle
Sphere
r
V=
2
2
15
R
3
A
P
B
Theorems
1. The volume of the prism is the product of the altitude and the area of the base.
2. The volume of a triangular pyramid is one – third the product of its altitude
and its base area.
3. The volume of pyramid is one-third the product of its altitude and its
base area.
4. The volume of a circular cylinder is the product of its altitude and the
area of its base.
Cylinder
5. The volume of a circular cone is one-third the product of its altitude and the area of its base.
Cone
V. ANALYTIC GEOMETRY
Introduction
A straight line is represented by an equation of the first degree in one or two variables, while circle, parabola,
ellipse and hyperbola are represented by equations of the second degree in two variables.
A. The Straight Line
1. The distance between two points A
2. Slope of a line
and B
is
a. The slope of the non-vertical line containing A
x1  x2 2   y1  y2 2 .
and B
is m 
y1  y 2
y  y1
or m  2
x1  x2
x2  x1
b. The slope of a line parallel to the x-axis is 0.
c. The slope of a line parallel to the y-axis is undefined.
d. The slope of a line that leans to the right is positive.
e. The slope of a line that leans to the left is negative.
3. The Equation of a line
In general, a line has an equation of the form ax + by + c = 0 where, a, b, c are numbers and that a and b
are not both zero.
4. Different forms of the equation of a line
a. General form:
b. Slope-intercept form:
, where is the slope and the y – intercept.
c. Point slope form:
where
is any point on the line.
d. Two point form: y  y1 
e. Intercept form:
y 2  y1
x  x1 where
x2  x1
and
are any two points on the line.
x y
  1 where a is the x-intercept and b the y-intercept.
a b
5. Parallel and Perpendicular lines
Given two non-vertical lines p and q so that p has slope m1 and q has slope m2.
a. If and are parallel, then
b. If and are perpendicular to each other, then
6. Segment division
Given segment AB with A
and B
.
 x1  x2 y1  y2 
,
.
2 
 2
r
AP r1

b. If a point P divides
in the ratio 1 so that
, then the coordinates of P
r2
PB r2
r x  r2 x1
ry r y
using the formula x  1 2
and y  1 2 2 1 .
r1  r2
r1  r2
a. The midpoint M of segment AB is M 
7. Distance of a point from a line
The distance of a point A
d
from the line
is given by
Ax1  By1  C
 A2  B 2
B. The Circle
16
can be obtained
1. Definition. A circle is the set of all points on a plane that are equidistant from a fixed point on the
plane. The fixed point is called the center, and the distance from the center to any point of the
circle is called the radius.
2. Equation of a circle
a) General form:
b) Center-radius form:
is equal to
where the center is at
and the radius
3. Line tangent to a circle
A line tangent to a circle touches the circle at exactly one point called the point of tangency. The
tangent line is perpendicular to the radius of the circle, at the point of tangency.
C. Conic Section
A conic section or simply conic, is defined as the graph of a second – degree equation in x and y.
In terms of locus of points, a conic is defined as the path of a point, which moves so that its distance
from a fixed point is in constant ratio to its distance from a fixed line. The fixed point is called the focus of
the conic, the fixed line is called the directrix of the conic, and the constant ratio is called the eccentricity,
usually denoted by .
If
If
If
, the conic is an ellipse. (Note that a circle has
, the conic is parabola.
, the conic is hyperbola.
)
D. The Parabola
1. Definition. A parabola is the set of all points on a plane that are equidistant from a fixed point and a
fixed line of the plane. The fixed point is called the focus and the fixed line is the directrix.
2. Equation and Graph of Parabola
a) The equation of a parabola with vertex at the origin and focus at
parabola opens to the right if
and opens to the left if
and opens downward if
c) The equation of a parabola with vertex at
. The
is
. The
.
b) The equation of a parabola with vertex at the origin and focus at
parabola opens upward if
is
.
and focus at
. The parabola opens to the right if
is
and opens to the left is
.
d) The equation of a parabola with vertex at
and focus at
. The parabola opens upward if
e) Standard form:
or
f)
, or
General form:
is
and opens downward is
.
3. Parts of a Parabola
a) The vertex is the point, midway between the focus and the directrix.
b) The axis of the parabola is the line containing the focus and perpendicular to the directrix.
The parabola is symmetric with respect to its axis.
c) The latus rectum is the chord drawn through the focus and parallel to the directrix (and
therefore perpendicular to the axis) of the parabola.
d) In the parabola
, the length of the latus rectum
is , and
the endpoints of the latus rectum are
and
y
.
O
In the figure at the right, the vertex of the parabola is the
origin, the focus is
, the directrix is the line
x
’
17
’
containing
, the axis is the x-axis, the latus rectum
is the line containing
.
y
y
’
x
O
x
O
’
The graph of
The graph of
Ellipse
1. Definition. An ellipse is the set of all points P on a plane such that the sum of the distances of P from
two fixed points F’ and F on the plane is constant. Each foxed points is called focus (plural: foci).
2. Equation of an Ellipse
a) If the center is at the origin, the vertices are at
, the foci are at
, the
endpoints of the minor axis are at
and
, then the equation
x2 y2
is 2  2  1 .
a
b
b) If the center is at the origin, the vertices are at
endpoints of the minor axis are at
and
is
, the foci are at
, the
, then the equation
x2 y2

1 .
b2 a 2
c) If the center is at
, the distance between the vertices is
horizontal and
, then the equation is
d) If the center is at
, the principal axis is
x  h 2   y  k 2
a2
b2
, the distance between the vertices is
y  k 
2
vertical and
, then the equation is
a2
1 .
, the principal axis is
2

x  h

b2
1 .
3. Parts of an Ellipse
For the terms described below, refer to the ellipse shown with center at O, vertices at
and
, foci at
and
, endpoints of the minor axis at
and

b2 

b2 
, endpoints of one latus rectum at G'   c,  and G  c,  and the other at
a
a



 b2 
b2 
H '  c,  and H  c,  .
a

 a
y

b2 
  c,


a 


 b2

 c, a





O

b2

  c, a

x

b2

 c, a









a) The center of an ellipse is the midpoint of the segment joining the two foci. It is the
intersection of the axes f the ellipse. In the figure above, point O is the center.
18
b) The principal axis of the ellipse is the line containing the foci and intersecting the ellipse at
its vertices. The major axis is a segment of the principal axis whose endpoints are the
vertices of the ellipse. In the figure,
is the major axis and has length
units.
c) The minor axis is the perpendicular bisector of the major axis and whose endpoints are
both on the ellipse. In the figure,
is the minor axis and has length
units.
d) The latus rectum is the chord through a focus and perpendicular to the major axis.
and
2b 2
are the latus rectum, each with a length of
.
a
y
9

  4, 
5

y
 9
 4, 
 5
x
O
9

  4, 
5

The graph of
x
O
5

 4, 
9

x2 y2

 1.
25 9
The graph of
x  22   y  12
100
25
1 .
4. Kinds of Ellipses
a) Horizontal ellipse. An ellipse is horizontal if its principal axis is horizontal. The graphs above
are both horizontal ellipses.
b) Vertical ellipse. An ellipse is vertical if its principal axis is vertical.
E. The Hyperbola
1. Definition. A hyperbola is the set of points on a plane such that the difference of the distances of each
point on the set from two fixed points on the plane is constant. Each of the fixed points is called focus.
2. Equation of a hyperbola
a) If the center is at the origin, the vertices are at
, the foci are at
, the
endpoints of the minor axis are at
and
, then the equation is
x2 y2

 1.
a2 b2
b) If the center is at the origin, the vertices are at
endpoints of the minor axis are at
, the foci are at
and
, the
, then the equation is
y2 x2

 1.
a2 b2
c) If the center is at
, the distance between the vertices is
horizontal and
d) If the center is at
, then the equation is
x  h 2   y  k 2
a2
, the distance between the vertices is
vertical and
, then the equation is
, the principal axis is
b2
, the principal axis is
 y  k 2  x  h 2
a2
1 .
b2
1 .
3. Parts of a hyperbola
For the terms described below, refer to the hyperbola shown which has its center at O,
vertices at
and
, foci at
and
, endpoints of one latus rectum at



 b2 
b2 
b2 
b2 
G'   c,  and G  c,  and the other at H '  c,  and H  c,  .
a
a
a



 a
y

b2

  c, a

 b2 
 c,


a 






x
O
19

b2

  c,  a






b2

 c,  a





a) The hyperbola consists of two separate parts called branches.
b) The two fixed points are called foci. In the figure, the foci are at
.
c) The line containing the two foci is called the principal axis. In the figure, the principal axis is
the x-axis.
d) The vertices of a hyperbola are the points of intersection of the hyperbola and the principal
axis. In the figure, the vertices are at
.
e) The segment whose endpoints are the vertices is called the transverse axis. In the figure
is the transverse axis.
f) The line segment with endpoints
and
where
is called the
conjugate axis, and is perpendicular bisector of the transverse axis.
g) The intersection of the two axes is the center of the hyperbola.
h) the chord through a focus and perpendicular to the transverse axis is called a latus rectum.
In the figure,

b2 
is a latus rectum whose endpoints are G '   c,  and
a


b2 
2b 2
.
G  c,  and has a length of
a
a

4. The Asymptotes of a Hyperbola
Shown in the figure below is a hyperbola with two lines as extended diagonals of the rectangle
shown.
y
b
x
a
y
y
b
x
a
x
These two diagonal lines are said to be the asymptotes of the curve, and are helpful in sketching the
b
x2 y2
 2  1 are y  x
2
a
a
b
2
2
b
a
y
x
and y   x . Similarly, the equations of the asymptotes associated with 2  2  1 are y  x
a
b
a
b
a
and y   x .
b
graph of a hyperbola. The equations of the asymptotes associated with
y
o
The graph of
y
x
o
x2 y 2

1.
9 27
The graph of
20
x
y2 x2

1 .
9 27
VI. CALCULUS
I.
Theorems on Limits of Functions
We use the following theorems to evaluate limits of functions:
1. Uniqueness Theorem: If the limit of a function exists, then it is unique. That is, if
, then
.
2. If
, then
3. If
, a constant, then
4.
5. If
and
then
5.1
5.2
5.3
provided
6. If
and
then
6.1
6.2
7.
with the restriction that if
is even,
.
if and only if
8. If is any positive integer, then
8.1
8.2
8.3
9. Let
,
and
where
9.1 If
and if
approaches 0 through positive values of
, then
9.2 If
and if
approaches 0 through negative values of
, then
9.3 If
and if
approaches 0 through positive values of
, then
9.4 If
and if
approaches 0 through negative values of
, then
10. Let
,
and
10.1
10.2
11. If
10.3
If
10.4
If
10.5
If
10.6
If
,
then
and
21
where is any constant, then
and
II.
Theorems on Differentiation
In getting the derivative of a function, one or more of the following theorems may be applied:
1. If
2. If
, where is a constant, then
, where
, then
3. If
, then
4. If
5. If
6. If
, then
, where is a constant, then
, where
, then
7. If is any rational number,
8. If
, then
9. If
, then
10. If
, then
11. If
, then
12. If
, then
13. If
, then
14. If
then
15. If
where
16. If
, then
is any constant not equal to 0, then
then
22
MODERN GEOMETRY & LINEAR ALGEBRA
MODERN GEOMETRY
Euclidean Geometry
The geometry with which are most familiar is called Euclidean geometry and is modeled by our notion of a “flat plane”
It was named after Euclid, a Greek Mathematician who lived in 300 BC. This geometry satisfies all Euclid’s postulates and
initial axioms namely:
Points Existence Postulate. Space contains at least four noncoplanar points. Every plane contains at least three noncollinear points. Every line contains at least two points.
Straight-Line Postulate. Two points are contained in one and only one line. (Two points determine a line.)
Plane Postulate. Three noncollinear points are contained in one and only one plane. (Three non collinear points determine
a plane.)
Flat Plane Postulate. If two points are in a plane, then the line containing the points is in the same plane.
Plane-intersection Postulate. It two planes intersect, then their intersection is a line.
Parallel Postulate. Through a given point P not on a line l, exactly one line may be drawn parallel to line l.
Euclidean Initial Axioms
1.
2.
3.
4.
5.
Any two distinct points determine one and only line.
Any three noncollinear points, or a line and a point not on the line, determine one and only one plane.
Any two distinct coplanar lines either intersect in one and only one point or are parallel.
Any line not in a given plane either intersects the plane in one and only one point or is parallel o the plane.
Any two distinct planes either intersect in one and only one line or are parallel.
Non-Euclidean Geometry
Non-Euclidean Geometry is any geometry that is different from Euclidean geometry. The two most common non-Euclidean
geometries are elliptic geometry and hyperbolic geometry/
A. Hyperbolic Geometry
Hyperbolic geometry is also known as saddle geometry or Lobachevskian geometry. It differs in many ways from Euclidean
geometry, often leading to quite counter-intuitive results. Some of the remarkable consequences of the geometry’s unique
fifth postulate include:
1. The sum of the three interior angles of a triangle is strictly less than 180⁰. Moreover, the angle sums of two distinct
triangles are not necessarily the same.
2. Two triangles with the same interior angles have the same area.
Also, in hyperbolic geometry more than one distinct line through a particular point will not intersect another given line.
B. Elliptic Geometry
In Elliptic geometry there are no lines that will not intersect, as all that start to separate will converge. For example, directly
from Euclidean geometry’s fifth postulate we have that there are no parallel lines. In particular, in elliptic geometry, given a
line and a point P not on , there are no lines through P parallel to
Some theorems in Elliptic Geometry
1. The angle sum of any triangle is more than 180⁰.
2. Given two lines perpendicular to line CG. By the parallel postulate for elliptic geometry, these two lines meet at
appoint A. Then every line through A is perpendicular to line CG.
23
C. Projective Geometry
Projective geometry is the most general and least restrictive in the hierarchy of fundamental geometries. It is an intrinsically
non-metric geometry, whose facts are independent of any metric structure. Under the projective transformations, the
incidence structure and the cross-ratio are preserved. In particular, it formalizes one of the central principles or perspective
art: that parallel lines meet at a point called an ideal point. Consequently, the five initial axioms in Euclidean geometry
resulted to the following axioms.
1.
2.
3.
4.
5.
Any two distinct points determine one and only one line.
Any three noncollinear points, also any line and a point not on the line, determine one and only one plane.
Any two distinct coplanar lines either intersect in one and only one point.
Any line not in a given plane either intersects the plane in one and only one point.
Any two distinct planes either intersect in one and only one line.
The following axioms serve as basis for constructing the plane projective geometry.
1.
2.
3.
4.
5.
If A and B are distinct points, then there is at least one line on both.
If A and B are distinct points, then there is at most one line on both.
If p and q are two distinct lines, then there is at least one point on both.
There are at least three distinct points on any line.
Not all points are on the same line.
MATRICES AND MATRIX OPERATIONS
Definitions
A matrix is defined is an rectangular array of elements. The entries, also called elements, may be real, complex or
function. If the arrangement has rows and columns, then the matrix is order of
(read as m by n). A matrix is
enclosed by a pair of parameters such as () or []. It is denoted by a capital letter.
Types of Matrices
1. The Row Matrix: This matrix has only one row.
Example:
[0 25 4 3]
This is a 1 x 4 row matrix
2. The Column Matrix: This matrix has only one column.
Example:
This is 4 x 1 column matrix
3. The Rectangular Matrix: This has two or more rows with two or more columns.
Example:
This is a 2 x 4 matrix, because it contains two rows and
four columns
This is a 3 x 4 matrix
4. The Square Matrix: This special case of a Rectangular Matrix; here the number of rows is equal to the
number of columns.
Example:
Here A and B are square matrices of order of 3 and 4 respectively
24
5. The Diagonal Matrix: This is a square matrix where all its non-diagonal elements are 0.
Example:
(a)
(b)
(C)
There are diagonal matrices or order 2, 3 and 4 respectively.
6. The Scalar Matrix: This is a diagonal matrix where all the elements on its leading diagonal to bottom
right are of equal value.
Example:
(a)
(b)
(C)
7. The Identity Matrix: This is a scalar matrix where the elements on its leading diagonal (the diagonal
running from top left to bottom right) are 1 and the rest are of value 0.
Example:
This is an identity matrix
of order 4.
Matrix Addition
Definition: If
and
are
matrices, then the sum of A and B is the
matrix
, defined by,
Properties of Matrix Addition
Theorem:
Let A, B, C, and D be matrices of the same size,
.
1. A + B = B + A
(Commutativity)
2. A + (B + C) = (A + B) + C
(Associativity)
3. There is a unique
matrix 0 such that A + 0 = A for any
called the
4. To each
matrix. The matrix 0 is
additive identity or zero matrix.
matrix A, there is a unique
matrix D such that
A+D=0
We shall write D as (-A), so that A + D = 0 can be written as A + (-A) = 0.
The matrix (-A) is called the additive inverse or negative of A.
Matrix Multiplication
Definition: If
denoted
is an
, is the
matrix and
matrix
is a
matrix, then the product of A and B,
, defined by,
Properties of Matrix Multiplication
Theorem:
Let A, B, and C be the matrices of the appropriate sizes.
1. A(BC) = (AB)C
(Associativity)
2. A(B + C) = AB + AC
(Right Distributivity)
3. (A + B)C = AC + BC
(Left Distributivity)
Scalar Multiplication
25
Definition: If
is an
matrix
That is,
matrix and is a real number, then the scalar multiple of A by ,
, where
is obtained by multiplying each element of
by
Properties of Multiplication by a Scalar
Theorem:
1.
2.
3.
4.
Let A and B be matrices of the appropriate sizes, and let r and s be scalars.
r(sA) = (rs)A
(Associativity)
(r + s)A = rA + sA
(Distributivity I)
r(A + B) = rA + rB
(Distributivity II)
A(rB) = r(AB) = (rA)B
Power of Matrices
If is a positive integer and
is a square (n x n) matrix, we define
factors
and
Theorem:
1.
2.
However, in general,
;
unless
.
Properties of the Transpose
Definition: Transpose of a Matrix. The matrix obtained from any given matrix , by interchanging the
rows and columns. Written as
(to be read as A – Transpose)
Example:
Theorem:
1.
2.
3.
4.
If r is a scalar and A and B are matrices, then
(AT)T = A
(A + B)T = AT + BT
(AB)T = BT AT
(rA)T = rAT
Remarks:
1. AB may not be equal to BA
2. AB = o does not imply that A = 0 or B = 0
3. AB = AC does not imply that B = C. They may be different.
Definition: A matrix A is called symmetric if, AT = A. That is, A is symmetric if it is a square matrix for
which
Example:
26
, is the
Remark: A matrix is symmetric if its entries are symmetric with respect to the main diagonal.
Definition: A matrix is called skew symmetric if
.
Example:
Remark: If A is a skew symmetric matrix, then the elements of the main diagonal
Definition: If
is an
matrix, then the trace of ,
.
, is defined as the sum of all elements on
the main diagonal of .
Determinants
Definition:
Let S = {1,2 … n} be the set of integers from 1 to n, arranged in ascending order. A
rearrangement
of the elements of S is called a permutation of S.
Definition:
A permutation
is said to have an inversion if a larger number
precedes a smaller number
Definition:
. If the total number of inversions in
is add then
is
called odd, if even then
is even.
Let
matrix. We define the determinant of A (written det( ) or
be an
)
by
where the summation ranges over all permutations
sign is taken as + or – according to whether the permutation
of the set S = {1,2 … n}. The
is even or odd.
Second – order Determinant
If A is the square matrix of order two
then the determinant of A, denoted by either det A or
, is defined by
Example: Compute the determinant:
Properties of Determinant
1. The determinant of a matrix and its transpose are equal, that is, det(AT) = det(A).
2. If two rows (columns) of A are equal, then det(A) = 0.
3. If a row (column) of A consists entirely of zeros, then det(A) = 0.
4. The determinant of a diagonal matrix is the product of the entries on its main diagonal.
5. If matrix
is upper (lower) triangular, then det(A) =
; that is the determinant of a
triangular matrix is the product of the elements on the main diagonal.
6. If a multiple of one row of A is added to another row to produce a matrix B, then det B = det A.
27
7. If two rows of A are interchanged to produce B, then det B = - det A.
8. If one row of A is multiplied by k to produce B, then det B = k det A.
9. The determinant of a product of two matrices is the product of their determinants; that is, det(AB) =
det(A) det(B).
10. If A is nonsingular, then det (A) ≠ 0 and det(A-1) =
28
.
ANALYZING TEST ITEMS
1. Find the GCF of 120, 80 and 140
a. 4
b. 12
c. 10
d. 20
2. When Joseph sorts his collection of marbles into groups of 2, 3, 4 or 8, there is always one marble left.
What is the smallest number of marbles Joseph can have?
a. 35
b. 28
c. 25
d. 21
3. Two brothers and their younger sister are to divide an inheritance worth P300, 000 n the ratio of 5:6:4,
with the girl getting the least share. How much is the share of the sister
a. P20, 000
b. P80,000
c. P100, 000
d. P40, 000
4. Which of the following statements is FALSE?
a. If a number is a multiple of 9, then it is also a multiple of 3.
b. If a number is divided by 9, then it is divisible by 3.
c. If 9 is a divisor of a number, then 3 is also a divisor of the number.
d. If a number is divisible by 3, then it is divided by 9.
5. Find
.
a.
b.
c. 4
d.
6. Which of the following numbers is the difference of two consecutive prime number less than 41?
a. 4
b. 6
c. 8
d. 9
7. Employees in a firm are entitled to 1 – day vacation leave for every 20 working days. If in 2007, this firm
had 280 working days, how many days of vacation leave are the employees entitled to?
a. 20 days
b. 15 days
c. 24 days
d. 14 days
8. What is the perimeter of a square whose area is 256 m2?
a. 1024 m
b. 64 m
c. 32 m
d. 16 m
9. How much bigger is 36 than 63?
a. 27
b. 235
d. 729
c. 513
10. Three brothers inherited a cash amount of P120, 000 and they divided it among themselves in the ratio of
5:2:1. How much more is the largest share than the smallest share?
a. P15, 000
b. P30, 000
c. P60, 000
d. P75, 000
11. The simplest expression
is ___________.
a.
12. 40% of 35 is what percent of 140?
a. 29%
b. 28%
b. 1
d. 220
c. 2
c. 10%
d. 14%
13. The initial temperature of an object was 27°C. After exposing the object to different surrounding media, its
temperature decreased by 7°C, then increased by 10°C and then finally decreased by 25°C. What was the
final temperature of the object?
a. 15°C
b. 20°C
c. -5°C
d. 5°C
14. A tank that holds 400 gallons of water can be filled by one pipe in 15 minutes and emptied by another in
40 minutes. How long would it take to fill the tank if both pipes are functioning?
a. 28 min
b. 21 min
c. 24 min
d. 23 min
29
15. Linda paid P360.00 for 12 notebooks but she was given 3 additional notebooks for free. In effect, what rate
of discount did she enjoy?
a. 50%
b. 40%
c. 18%
d. 20%
16. A clock is 4 minutes ahead every 8 hours. If the clock is set correctly at 8:00 a.m. Monday, what time will
be shown in this clock at 8:00 p.m. Friday?
a. 8:48 p.m.
b. 8:54 p.m.
c. 8:56 p.m.
d. 9:08 p.m.
17. What is the LEAST positive integer that has 6, 8, and 10 as factors?
a. 240
b. 80
c. 120
d. 300
18. Jose walks M miles in H hours. At the same rate how many miles will he walk in J hours?
a.
b.
19. What number is NOT exactly divisible by 8?
a. 7304
b. 5000
c.
d.
c. 3584
d. 5218
20. Nine bus stops are equally spaced along a bus route. The distance from the first to the third stop is 600
meters. How far is the distance form the first stop to the last stop?
a. 2100 meters
b. 2400 meters
c. 900 meters
d. 2700 meters
21. Three out of every 24 flashlights turned out by a particular factory are found to be defective. If the factory
turns out 1248 flashlights in a week, how many are defective?
a. 72
b. 18
c. 416
d. 156
22. If ab + cd = 12 ad, and ad ≠ 0, then
a. ab + cd
.
b. 12
c. 12a + 12d
d. 12 ac
23. What must be the value of x in the arithmetic progression x – 7, x – 2, x + 3 so that its 10th term will be 40?
a. 3
b. 2
c. 1
d. 4
24. The roots of the equation (x – 5)(x – 3) = 0 are _____.
a. 3 or -5
b. 3 or 5
c. -3 or 5
d. -3 or -5
25. The average of m and n is 9, and p = 12. What is the average of m, n, and p?
a. 7
b. 21
c. 10
d. 18
26. An angle of
radians is equal to _______.
a. 115°
b. 225°
27. What is the value of x in
a. 2
c. 105°
d. 89°
c. 1
d. 4
?
b. 3
28. M varies inversely as N. If M = 25 when N = 2, what is M when N = 5?
a. 16
b. 10
c. 5
d. 2
29. Which of the following expression always gives an odd number?
a. (N – 1)(N – 5)
b. 5N – 1
c. (2N + 1)2
d. N2 – 1
30. Three circles are tangent to each other externally. What is the perimeter of the triangle formed by
connecting the centers if the areas of the circles are 9 , 16 and 25 respectively?
a. 24 cm.
b. 12 cm.
c. 50cm.
d. 25 cm.
30
31. A falling body strikes to the ground with a velocity V, which varies directly as the square root of the
distance d it falls. If a body that falls 100 feet strikes the ground with a velocity of 80 ft./sec., with what
velocity will a ball dropped from a height of 550 ft. strike the ground?
a. 150 ft./sec.
b. 190 ft./sec.
c. 170 ft./sec.
d. 188 ft./sec.
32. Which of the following is a true identity?
a. sin2θ = cos2θ
b. cos2θ + 1 = sin2θ
c. tan2θ = sec2θ – 1
d. cos2θ – 1 = sin2θ
33. The same number is subtracted from the numerator and denominator of the fraction
. If the resulting
fraction is equivalent to , what is the number subtracted?
a. 5
34. What is
b. 2
c. 3
d. 4
b. 4
c. 16
d. 6
?
a. 5.4
35. Given this table of numbers relating x to y, what is x when y is 45?
x
0
4
8
12
16
y
3
9
15
21
27
a. 22
b. 28
c. 48
d. 46
36. Madel spent one-sixth of her money in one store. In the next store, she spent three times as much as she
spent I the first store, and had P80.00 left. How much money did she have at the start?
a. P240.00
b. P360.00
c. P252.00
d. P380.00
37. Which gives the quantity “the time it takes to read a book that is x page long at a rate of y pages per
hour”?
a. hours
b. xy hours
c. hours
d. (x + y) hours
. then 16y2 equals _________.
38. If
a.
b.
c. x3z
d. x2z2
39. Marlin copied x + y2 instead of (x + y)2 . Compute the amount of her error if x = 8 and y = 3.
a. 12
b. 0
c. 104
d. 96
40. Which statement is TRUE?
I.
1m is longer than 1 km
II.
1mm is longer than 1 in.
III.
1 gallon is heavier than 1 lb.
IV.
1 gallon is more than 1 liter
a. I only
b. I and IV
41. If 3x = 81 and 4y = 64. What is (x – y)?
a. 17
b. 1
c. I and III
d. III and IV
c. 0
d. -1
42. Two buses leave the same station at 9:00 p.m. ones bus travels north at the rate of 30 kph and the other
travels east at 40 kph. How many kilometers apart are the buses at 10 p.m.?
a. 100 km
b. 70 km
c. 50 km
d. 140 km
43. Which of the following is a factor of x4 – 4x3 – 6x2 + 3x + 10?
a. (x – 2)
b. (x + 5)
c. (x + 1)
d. (x – 5)
44. If X is an even number and y is odd number, which of theses expressions will always give an even number?
a. X + Y
b. XY
c. X ÷ Y
d. X – Y
31
45. Which of these is NOT factorable?
a. 6x2 – 5x – 6
b. 6x2 – 13x + 6
c. 6x2 + 7x – 6
d. 6x2 – 35x – 6
c. 1
d. 3
46. What is the value of x in this equation?
a. 4
b. 2
47. If 16 is 4 mores than 3x, then what is the value of 2x – 5?
a. 2
b4
c. 5
d. 3
48. What is the area of the shaded portion as shown in the figure?
4’
2’
a. 8 -
b.
c. 8 - 4
d. 8 - 2
49. Which on these could be the measure of the angles of an isosceles triangle?
a. 60°, 60°, 80°
b. 32°, 32°, 116°
c. 51°, 98°, 51°
d. 45°, 45°,100°
50. A box is 12 in. in width, 16 in. in length, and 6 in. height. How many square inches of paper would be
required to cover it on all sides?
a. 900
b. 192
c. 720
d. 360
51. What is the total length of fencing needed to enclose a rectangular area 46 feet by 34 feet?
a. 48 yd.
b. 26 yd. 1 ft.
c. 53 yd.
d. 52 yd. 2 ft.
52. If the area of a square is 81x2, what will represent the perimeter of the square?
a. 18x2
b. 81x
c. 9x2
d. 36x
53. The vertices of a triangle are (2, 1), (2, 5), and (5, 1). What is the area of a triangle?
a. 10
b. 8
c. 5
d. 6
54. Four small squares are put together to form a bigger square. If the perimeter of the big square is 256, what
is the side of each of the smaller squares?
a. 24
b. 16
c. 32
d. 8
55. Three segments have lengths a, b and c with a < b < c. Under what conditions these segments could be
used to for a triangle?
a. a + b > b
b. b + c > a
c. a + c < b
d. a + b > c
56. The measure of each interior angle of a regular polygon is 165 degrees. How many sides does it have?
a. 12
b.15
c. 24
d. 29
57. What happens to the circumference of a circle if the length of the radius is doubled?
a. It becomes 3 times longer.
c. It is doubled.
b. It becomes 4 times longer.
d. It is halved.
58. Which of the following equation is parallel to 3x – 2y = 4 and has a y – intercept of -3?
a. y = 2x – 3
b. 2x – 3y = 6
c. 2x – 3y = 4
d. 3x – 2y = 6
59. All of the quadrilaterals have two pairs of opposite sides parallel EXCEPT
a. Rectangle
b. Trapezoid
c. Square
32
d. Rhombus
60. The area of a circle is 49 . What is its circumference, in terms of ?
a. 49
b. 14
c. 28
d. 98
61. Which line in the figure has an undefined slope?
1
a. 4
2
3
b. 3
4
c. 2
d. 1
62. What is the slope of the line passes through (-1, 2) and (3, -4)?
a.
b. 3
c. 2
d.
63. A rectangular bin 4 feet long, 3 feet wide and 2 feet high is solidly packed with bricks whose dimensions
are 8 inches, 4 inches and 2 inches. How many bricks are there in the bin?
a. 54
b. 848
c. 648
d. 320
64. P is 7 less than the square of the sum of the quotient of x and y and the product of x and y. which equation
express this relationship?
a.
b.
c.
d.
b.
c.
d.
65. What is 135° in radians?
a.
66. A square is inscribed in a circle with radius r. Find the area of the shaded region.
r
a. (2 - )r2
b. ( - 2)r2
c.
- 2r2
d. r2 - 2
67. If the first derivative of f(x) is 3x2 + 2x, what is f(x)?
a. 3x3 + 2x
b. x3 + x2
c. 6x3 + 2x2
d. 3x3 + x
68. What are the endpoints of the major axis of 9(x – 3)2 + 25(y – 2)2 = 225?
a. (1, -2) and (-7, -2)
b. (0, 2) and (6, 2)
C. (-1, 2) and (7, 2)
d. (-2, 2) and (8, 2)
69. What is the equation of a straight line with a gradient -1 and y – intercept 4?
a. 2y = x + 4
b. 4y = x + 1
c. y = 2x + 5
d. y = -x + 4
70. If f(x) = 5x3 – x2 + 3, find the second derivative of f(x).
a. 45x2 – 2x
b. 30x – 2
c. 3x2 – 2x
71. Evaluate:
a. -4
d. 15x3 – 2xy
is _________.
b. 4
72. What is the second derivative of y = (x3 + 2)2?
a. 2x2(x3 – 2)
b. 6x(x3 + 2)
c. 2
d.
c. 6x(5x3 +4)
d. 6x2(5x2 + 8)
73. The cost C(x) in pesos of producing x math textbooks is given by
the cost of each book?
a. P11.50
b. P20.00
c. P31.50
33
. How much is
d. P230.00
74. Determine the equation of the parallel to 3x – 8y + 11 = 0 and passing through (8, 10).
a. 8x – 3y – 34 = 0
b. 3x – 8y + 56 = 0
c. 3x – 8y – 104 = 0
d. 3x + 8y + 56 = 0
75. What is the equation of the line with an x – intercept of 5 and a y – intercept of 27?
a. 5y + 2x = 4
b. 2y + 5x = 25
c. 2y + 5x = 10
d. 5y + 2x = 10
76. A particular car runs 100 km on x liters of gasoline. If the car travels a fixed route of 20 km. per day, how
many liters of gasoline will the car consume in 365 days?
a. 73x
b.
c. 36x
d.
77. One end of a line segment is P(1, -2). If the midpoint of the segment is (4, 3), find the coordinate of the
other endpoint of the segment.
a. (6, 9)
b. (6, 7)
c. (5, 8)
d. (7, 8)
78. What is the equation of the line through (4, 1) and parallel to the line through (7, 3) and (5, -1)?
a. x – 4y = 0
b. 3x + y = 13
c. 2x – y = 7
d. 5x – 6y = 14
79. What is the equation of a circle with center (-5, 7) and a diameter that is 20 cm?
a. (x + 5)2 + (y – 7)2
c. (x – 5)2 + (y + 7)2
b. (x – 5)2 + (x + 7)2 = 100
d. (x + 5)2 + (x – 7)2 = 100
80. Find the first derivative of (x2 + 2x)(x3)
a. x4 + 6x3
b.2x4 + 2x3
81. What is the equation of the line with slope
a. 6x + 11y – 75 = 0
c. 5x4 + 8x3
d. 3x4 + 6x3
and passes through C(4, 9)?
b. 11x + 6y + 10 = 0
c. 6x – 11y – 123 = 0
d. 6x + 11y – 123 = 0
82. How long is the diameter of x2 + y2 – 10x + 14y = -10?
a. 128 units
b. 16 units
c. 8 units
d. 64 units
83. In the graph of y = 2 – 2x2 – 4x, what is the highest point?
a. (1, -4)
b. (4, -1)
c. (-4, 1)
d. (-1, 4)
84. What are the coordinates of the point of intersection of the tangent with slope equal to 2 the curve y = -x2
– 4x + 7?
a. (-3,13)
b. (-3, 28)
c. (-3, 10)
d. (3, 4)
85. The derivative of y = 2x3 – 5x2 + 3x + 2 is _________________.
a. y1 = 6x2 – 10x – 3
b. y1 = 6x2 – 6x + 3
c. y1 = -6x2 + 10x – 3
d. y1 = 6x2 – 10x + 3
86. Which of the following quadratic functions will have a graph which opens downward?
a. y = 1 – (3x – x2)
b. y = 5x(x – 1) + 3
c. y = 7x – (5x2 + 1)
d. y = -4(2 – x2) + 5x
87. What is the median of the following set of numbers: 11, 8, 3, 2, 3.2, 14, 13, 8, 7.8, 6?
a. 8.2
b. 7.90
c. 7.45
d. 5
88. The scores of eight boys in an IQ TEST are as follows: 100, 120, 101, 107, 115, 116, 122 and 101. What is
the 75 percentile?
a. 120
b. 118
c. 116
d. 117
89. A pair of dice is thrown. If the sum of the dots o the two dice is 6, what is the probability that one of the
dice id 2?
a.
b.
c.
d.
90. Determine the length of the latus rectum of the parabola: 5y2 + 20y – 6x + 20 = 0
34
a. 6
b.
c.
d.
91. If majority of the students in Statistics class was able to answer of the questions given during the
examination, the score is expected to be __________.
a. bimodal
b. normally distributed c. negatively skewed d. positively skewed
92. Thirty people were asked a question answerable of yes, no and maybe, their responses were recorded.
Which measure of central tendency is MOST appropriate for the recorded data?
a. Range
b. Mode
c. Mean
d. Median
93. Francisco received the following scores on tests in Calculus: 73, 89, 81, 95 and 87. What is his mean score?
a. 84
b. 85
c. 89
d. 87
94. How many ways can 3 books be selected from a group of 7 books?
a. 42
b. 35
c. 210
d. 21
95. The president of the senior class is authorized to appoint a committee of 5 from 4 elected males and 8
elected females. How many different committees are possible if 3 elected males re to be members?
a. 32
b. 112
c. 120
d. 220
96. How many different signals, each consisting of 7 flags hung in a vertical line, can be formed fro 4 identical
red flags and 3 identical green flags?
a.
b. (7 – 4)!
c. 7!
d. 4!3!
97. How many distinct permutations can be made for the letters o the word INFINITY if all the letters are to be
taken?
a. 3540
b. 4512
c. 3360
d. 3124
98. Four married couples have reserved eight seats in a row for a basketball game. In how many ways can they
be seated if each couple is to sit together?
a. 350
b. 384
c. 320
d. 300
99. A family has 3 children. What is the probability that all children are boys?
a.
b.
c.
d.
100. A card is selected at random from a box of 20 cards numbered 1 to 20. What is the probability that the
number is multiple of 3?
a.
101.
b.
c.
d.
How much interest would be collected on a loan of P25, 000 borrowed for 7 months at 10% per year?
a. P1,485.33
b. P1,548.33
c. P1,458.33
d. P1,584.33
102. A store is giving a 15% discount off the marked price of electrical items, what is the discounted price of
a water dispenser marked at P3,000?
a. P3,150
b. P2,550
c. P2,850
d. P3,450
103.
If
, what is
a.
?
b.
c.
d.
104.
In a sale, Janice paid P270.00 for a skirt originally marked P450.00. What was the discount rate given?
a. 20%
b. 60%
c. 50%
d. 40%
105.
If
a.
, and
, find .
b.
c.
35
d.
106. Atty. Bravo collected P10,750 from Mrs. Rivera. After deducting the commission, the principal received
P8,492.35. What was the attorney’s rate of commission?
a. 27%
b. 12%
c. 21%
d. 79%
107.
In one sophomore class, of the students are honor students, of them are varsity athletes. If there
are four honor-athlete students in the class, how many students are there in the class?
a. 42
b. 54
c. 73
d. 62
108.
How much money was borrowed at 15% simple interest for 8 months if the interest paid was P3,500?
a. P36,500
b. P36,000
c. P35,000
d. P37, 200
109. Which of theses numbers CANNOT be the hypotenuse of a right triangle whose sides are whole
numbers?
a. 10
b. 13
c. 26
d. 4
110. A man sold his car for P180, 000. This means losing 10% of the price he paid fir it. What was the cost of
the car when he bought it?
a. P195,000
b. P190,000
c. P200,000
d.P210,000
111.
What is the area of a square lot whose diagonal is
a. 28 sq. m.
112.
b. 72 sq. m.
Find the value of
a.
if
m?
c. 12 sq. m.
d. 36 sq. m.
c. 1
d.
and
b.
113. A businessman borrowed P50,000 from a rural bank at 14% compound interest, compounded semi –
annually, how much did he pay at the end of two years?
a. P61,252.15
b. P57,245
c. P53,407.15
d. P65,539.80
114. An equilateral triangle is drawn with one vertex at the origin and the two other vertices on the circle x2
+ y2 = 25. Find the area of the triangle.
a.
b.
c.
d. 25
115. Jocelyn has P7,200 in the time deposit I a bank which she receives 12% interest each year. How much
interest will she receive in five years?
a. P7,200
b. P4,320
c. P4,800
d. P5,200
116. The interest of a loan of P6,350.00 is P158.75. If the loan is to be paid after 150 days, what is the rate
of interest charged?
a. 5%
b. 7%
c. 6%
d. 4%
117.
Which of these is equal to
a.
118.
119.
b.
What is the second derivative of y = (x3 + 2)2?
a. 6(x5 + 3x2)
b. 2x2 (x3 + 2)
c.
d.
c. 6x2 (x3 + 2)
d. 6x(x3 + 2)
Transform an angle of 190 degrees to radian measure.
a.
120.
?
If
a.
radians
, what is
b.
radians
c. 3.39 radians
d.
c.
d. 1
?
b. 2
36
radians